关于亚直既约环的一些有限条件

Subdirectly Irreducible Rings Where Some Finite Conditions are Satisfied

摘要: 证明了具有幂等心H的亚直既约环R若满足下列条件之一:(1)H只有有限个非零幂零元;(2)H只有有限个非零元x:x_k=0(某正整数k>1);(3)H只有有限个非零元x:x₂=0;(4)H只有有限个非零右零因子;(5)H只有有限个非零左零因子;(6)H只有有限个不为单位元的非零幂等元,则R≌M_n(F)(n≥2),其中F=GF(p_m)。

Abstract: It is proved that if a subdirectly irreducible ring R with the heart H=H² satisfies one of the following of conditions,then R≌M_n(F)(n≥2),where F=GF(P^m).(1) only finite non-zero nilpotent elements in H;(2) Only finite non-zero elements x in H,such as x^k=0;(where k>1);(3) only finite non-zero elements x in H, such as x²=0;(4) only finite non-zero right annihilators in H;(5) only finite non-zero left annilihilators in H;(6) only finite non-zero idmpotents in H,where e≠1.